Introduction to particle-in-cell simulation
Classical plasma physics considers the motion of charged particles. The dynamics of these particles will be effected by the presence of externally imposed electric and magnetic fields, which is relatively easy to model since the motion of each particle is independent. However, the particles will themselves source an electric field–-and, if they are moving fast enough, a magnetic field–-due to their charge. The dynamics of other particles will be influenced by these 'self-consistent' fields, which corresponding source fields of their own. Thus, the dynamics of all of the particles is coupled, and so their equations of motion must be solve together. For a typical plasma, the resulting differential equations cannot be solved analytically, and the number of degrees of freedom means that direct computational integration of the equations is also impossible. Thus, plasma physics relies on various simplifications and assumptions to create reduced models that can be solved–- either exactly, or approximately.
Approximations to the true particle distribution function
It is convenient to represent the locations and momenta of the particles using the distribution function
\[f(x,p,t) = \sum_{i=1}^N \delta(x - x_i(t)) \delta(p - p_i(t)),\]
where $N$ is the total number of particles. The evolution of this distribution function can be written using the Maxwell-Boltzmann system
\[\begin{aligned} TODO TODO-Maxwell \end{aligned}\]
Unfortunately, for almost all plasmas of interest, $N$ is enormous; thus direct simulation of the equations of motion is computationally intractable. The solution is to recognize that if $N$ is large, then any physically small phase-space region will contain many particles, and so we c
Todo
- course-graining
- collisionless plasmas
- briefly mention thermalization, two-fluid, and MHD
Discretized solutions of the Boltzmann-Poisson equations
We have seen that a kinetic plasma can be approximated using the Vlasov-Boltzmann equation, along with an appropriate equation of motion for the electromagnetic fields. However, the resulting equation is still not easy to analyse for an arbitrary $f$. We therefore turn to computational simulation of the equations of motion to understand the plasma dynamics.
In order to simulate the plasma, we must first discretize the equations so they can be represented on a computer. One option, called a Vlasov code, represents the distribution function as